Theorem zfcndext 4888

Description: Axiom of Extensionality, reproved from conditionless ZFC version and predicate calculus.

Assertion

Ref	Expression
zfcndext	|- (A.z(z e. x <-> z e. y) -> x = y)

Distinct variable group:   x,y,z

Proof of Theorem zfcndext

Step	Hyp	Ref	Expression
1		axextnd 4866	. . 3 |- E.z((z e. x <-> z e. y) -> x = y)
2	1	19.35i 1052	. 2 |- (A.z(z e. x <-> z e. y) -> E.z x = y)
3		19.9v 1266	. 2 |- (E.z x = y <-> x = y)
4	2, 3	sylib 198	1 |- (A.z(z e. x <-> z e. y) -> x = y)

Syntax hints:   -> wi 3   <-> wb 146  A.wal 950  E.wex 956   = wceq 1099   e. wcel 1105

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-13 1107  ax-14 1108  ax-17 1190  ax-ext 1436

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 957

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